For the béarnaise sauce:
Fill a large saucepan a quarter of the way full with water and bring it to a simmer over high heat. Remove from the heat and set aside.
Combine the wine, vinegar, shallot, tarragon stems, and crushed peppercorns in a medium frying pan. Bring to a simmer over medium heat and cook until reduced to just over 2 tablespoons, about 5 to 7 minutes.
Strain the vinegar mixture through a fine-mesh strainer into a small bowl, pressing on the solids; discard the solids. Set aside to cool, about 10 minutes.
Place the egg yolks and the cooled vinegar reduction in a blender and blend on medium speed until the mixture lightens in color, about 30 seconds.
Reduce the blender speed to low, remove the small cap from the lid (the pour lid), and slowly drizzle in all of the melted butter.
Turn off the blender; add the chopped tarragon leaves, chervil (if using), salt, and pepper; and pulse to combine.
To keep the béarnaise sauce warm, place the blender pitcher directly in the reserved hot water while you cook the steak and poach the eggs.
For the steak:
Heat a medium heavy-bottomed frying pan (not nonstick) over high heat until very hot but not smoking, about 3 to 4 minutes.
Meanwhile, season the steak on both sides with the measured salt and pepper. Rub both sides with the olive oil.
Place the steak in the hot pan and let it cook undisturbed until a dark crust forms on the bottom, about 3 to 4 minutes.
Flip the steak using tongs or a spatula and cook until it’s medium rare, about 3 to 4 minutes more.
To check for doneness, use your finger to press on the steak: It should be firm around the edges but still give in the center. You can also use an instant-read thermometer—it should read 125°F to 130°F.
Transfer the steak to a cutting board and let it rest while you poach the eggs.
For the poached eggs:
Bring the measured water to a simmer in a medium pot over high heat (the water should be at least 2 inches deep). Reduce the heat to low to keep the water at a bare simmer. Stir in the vinegar.
Break 1 egg into a small cup or ramekin. Gently slide the egg into the simmering water.
Repeat with the remaining 3 eggs, making sure to evenly space them in the water.
Cook until the whites are just set, about 3 minutes.
Lift the eggs out of the water with a slotted spoon and place in a bowl of warm water to keep warm.
Split and toast the English muffins. Coat the cut sides of the muffins with a thin layer of butter, season with salt and pepper, and place on serving plates.
Thinly slice the steak and divide it among the muffin halves.
Using a slotted spoon, remove the eggs from the warm water.
Blot any excess water from the underside of the spoon with a paper towel or kitchen towel and slide the eggs onto the steak.
Top the eggs with the desired amount of béarnaise sauce (you may have some left over to serve on the side).
Serve immediately and enjoy!